Proof of the Isoperimetric Theorem in Higher Dimensions
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I have read a couple of nice proofs for the isoperimetric theorem in 2 dimensions. Is there a simple proof for the isoperimetric theorem in $n$ dimensions? In other words, how do you prove that the $n$-dimensional "sphere" is the geometric figure with the largest "volume" with a given "area"?
calculus geometry
asked Jul 20 '16 at 23:23
HrhmHrhm
2,183417
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add a comment |
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I have read a couple of nice proofs for the isoperimetric theorem in 2 dimensions. Is there a simple proof for the isoperimetric theorem in $n$ dimensions? In other words, how do you prove that the $n$-dimensional "sphere" is the geometric figure with the largest "volume" with a given "area"?
calculus geometry
asked Jul 20 '16 at 23:23
HrhmHrhm
2,183417
$endgroup$
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www.math.utah.edu/~treiberg/Steiner/SteinerSlides.pdf
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– Jack D'Aurizio
Jul 21 '16 at 0:52
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that $dS$ an infinitesimal piece of sphere is the minimizer of $min_Sfrac{area(S)}{volume(convexHull(S,c))}$ (where $c$ is the center of the sphere) ?
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– reuns
Jul 21 '16 at 1:01
add a comment |
$begingroup$
I have read a couple of nice proofs for the isoperimetric theorem in 2 dimensions. Is there a simple proof for the isoperimetric theorem in $n$ dimensions? In other words, how do you prove that the $n$-dimensional "sphere" is the geometric figure with the largest "volume" with a given "area"?
calculus geometry
asked Jul 20 '16 at 23:23
HrhmHrhm
2,183417
$endgroup$
I have read a couple of nice proofs for the isoperimetric theorem in 2 dimensions. Is there a simple proof for the isoperimetric theorem in $n$ dimensions? In other words, how do you prove that the $n$-dimensional "sphere" is the geometric figure with the largest "volume" with a given "area"?
calculus geometry
calculus geometry
asked Jul 20 '16 at 23:23
HrhmHrhm
2,183417
asked Jul 20 '16 at 23:23
HrhmHrhm
2,183417
asked Jul 20 '16 at 23:23
HrhmHrhm
2,183417
asked Jul 20 '16 at 23:23
HrhmHrhm
2,183417
asked Jul 20 '16 at 23:23
HrhmHrhm
2,183417
2,183417
$begingroup$
www.math.utah.edu/~treiberg/Steiner/SteinerSlides.pdf
$endgroup$
– Jack D'Aurizio
Jul 21 '16 at 0:52
$begingroup$
that $dS$ an infinitesimal piece of sphere is the minimizer of $min_Sfrac{area(S)}{volume(convexHull(S,c))}$ (where $c$ is the center of the sphere) ?
$endgroup$
– reuns
Jul 21 '16 at 1:01
add a comment |
$begingroup$
www.math.utah.edu/~treiberg/Steiner/SteinerSlides.pdf
$endgroup$
– Jack D'Aurizio
Jul 21 '16 at 0:52
$begingroup$
that $dS$ an infinitesimal piece of sphere is the minimizer of $min_Sfrac{area(S)}{volume(convexHull(S,c))}$ (where $c$ is the center of the sphere) ?
$endgroup$
– reuns
Jul 21 '16 at 1:01
$begingroup$
www.math.utah.edu/~treiberg/Steiner/SteinerSlides.pdf
$endgroup$
– Jack D'Aurizio
Jul 21 '16 at 0:52
$begingroup$
www.math.utah.edu/~treiberg/Steiner/SteinerSlides.pdf
$endgroup$
– Jack D'Aurizio
Jul 21 '16 at 0:52
$begingroup$
that $dS$ an infinitesimal piece of sphere is the minimizer of $min_Sfrac{area(S)}{volume(convexHull(S,c))}$ (where $c$ is the center of the sphere) ?
$endgroup$
– reuns
Jul 21 '16 at 1:01
$begingroup$
that $dS$ an infinitesimal piece of sphere is the minimizer of $min_Sfrac{area(S)}{volume(convexHull(S,c))}$ (where $c$ is the center of the sphere) ?
$endgroup$
– reuns
Jul 21 '16 at 1:01
add a comment |
1 Answer
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The proof of the isoperimetric inequality in higher dimensions follows easily from the Brunn-Minkowski inequality. This approach is particularly well suited to bodies with non-smooth boundaries and also works for non-Euclidean geometries.
Self-contained proofs that do not rely on Brunn-Minkowski were given for bodies with smooth boundaries by Schmidt and Dilts, for compact Lebesgue measurable sets by
Hadwiger and for sets of finite perimeter by de Giorgi.
answered Feb 18 '17 at 13:44
Jan Peter SchäfermeyerJan Peter Schäfermeyer
1916
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1 Answer
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$begingroup$
The proof of the isoperimetric inequality in higher dimensions follows easily from the Brunn-Minkowski inequality. This approach is particularly well suited to bodies with non-smooth boundaries and also works for non-Euclidean geometries.
Self-contained proofs that do not rely on Brunn-Minkowski were given for bodies with smooth boundaries by Schmidt and Dilts, for compact Lebesgue measurable sets by
Hadwiger and for sets of finite perimeter by de Giorgi.
answered Feb 18 '17 at 13:44
Jan Peter SchäfermeyerJan Peter Schäfermeyer
1916
$endgroup$
add a comment |
$begingroup$
The proof of the isoperimetric inequality in higher dimensions follows easily from the Brunn-Minkowski inequality. This approach is particularly well suited to bodies with non-smooth boundaries and also works for non-Euclidean geometries.
Self-contained proofs that do not rely on Brunn-Minkowski were given for bodies with smooth boundaries by Schmidt and Dilts, for compact Lebesgue measurable sets by
Hadwiger and for sets of finite perimeter by de Giorgi.
answered Feb 18 '17 at 13:44
Jan Peter SchäfermeyerJan Peter Schäfermeyer
1916
$endgroup$
add a comment |
$begingroup$
The proof of the isoperimetric inequality in higher dimensions follows easily from the Brunn-Minkowski inequality. This approach is particularly well suited to bodies with non-smooth boundaries and also works for non-Euclidean geometries.
Self-contained proofs that do not rely on Brunn-Minkowski were given for bodies with smooth boundaries by Schmidt and Dilts, for compact Lebesgue measurable sets by
Hadwiger and for sets of finite perimeter by de Giorgi.
answered Feb 18 '17 at 13:44
Jan Peter SchäfermeyerJan Peter Schäfermeyer
1916
$endgroup$
The proof of the isoperimetric inequality in higher dimensions follows easily from the Brunn-Minkowski inequality. This approach is particularly well suited to bodies with non-smooth boundaries and also works for non-Euclidean geometries.
Self-contained proofs that do not rely on Brunn-Minkowski were given for bodies with smooth boundaries by Schmidt and Dilts, for compact Lebesgue measurable sets by
Hadwiger and for sets of finite perimeter by de Giorgi.
answered Feb 18 '17 at 13:44
Jan Peter SchäfermeyerJan Peter Schäfermeyer
1916
edited Jan 11 at 13:48
answered Feb 18 '17 at 13:44
Jan Peter SchäfermeyerJan Peter Schäfermeyer
1916
answered Feb 18 '17 at 13:44
Jan Peter SchäfermeyerJan Peter Schäfermeyer
1916
answered Feb 18 '17 at 13:44
Jan Peter SchäfermeyerJan Peter Schäfermeyer
1916
1916
add a comment |
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$begingroup$
www.math.utah.edu/~treiberg/Steiner/SteinerSlides.pdf
$endgroup$
– Jack D'Aurizio
Jul 21 '16 at 0:52
$begingroup$
that $dS$ an infinitesimal piece of sphere is the minimizer of $min_Sfrac{area(S)}{volume(convexHull(S,c))}$ (where $c$ is the center of the sphere) ?
$endgroup$
– reuns
Jul 21 '16 at 1:01